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The Hawking effect and the bounds on greybody factor for higher dimensional Schwarzschild black holes

In this work, we have considered a $n-$dimensional Schwarzschild-Tangherlini black hole spacetime with massless minimally coupled free scalar fields in its bulk and $3-$brane. The bulk scalar field equation is separable using the higher dimensional spherical harmonics on $(n-2)-$sphere. First, using the Hamiltonian formulation with the help of the recently introduced near-null coordinates we have obtained the expected temperature of the Hawking effect, identical for both bulk and brane localized scalar fields. Second, it is known that the spectrum of the Hawking effect as seen at asymptotic future does not correspond to a perfect black body and it is properly represented by a greybody distribution. We have calculated the bounds on this greybody factor for the scalar field in both bulk and $3-$brane. Furthermore, we have shown that these bounds predict a decrease in the greybody factor as the spacetime dimensionality $n$ increases and suggest that for a large number of extra dimensions the Hawking quanta is mostly emitted in the brane.